# Logic and Set Theory/Implications, equivalences

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## Implications[edit]

We begin by introducing a fundamental relation in logic, the implication.

To do so, we first define what a logical statement is. A logical statement dependent on several *variables*, which we denote by , is a statement, which, dependent on what values take, is either true or false. For example, consider the statement

- .

Then we will have that is true, and is false; indeed, , but certainly not .

With this in mind, we may define the implication.

**Definition (implication)**:

Let , be logical statements dependent on . Saying that implies (written ) means that whenever is true for given , then is also true.